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saw activity at simple object and started a tiny section with examples.
Looking at this article: what is the need for having a zero object as opposed to just a terminal? In other words, would it be worse to say an object is simple if it has precisely two quotients, itself and $1$?
I’m particularly interested in the case of rings and commutative rings. One definition of simple ring is that it has precisely two two-sided ideals, and this condition is equivalent to the one I’m proposing. Similarly, a simple commutative ring under my proposal is just a field.
(It may be that the concept becomes interesting only under additional assumptions, such as Barr-exactness or something in that vein. That’s why I phrased my question as I did: would my proposal be any worse than the one given?)
The definition of simple ring on the nlab is $R$ is simple as a bimodule. This seems to be equivalent to your definition. As far as I can tell your definition should subsume that definition of simple ring. I think the terminology becomes problematic for Lie algebras however.
Yeah, for Lie algebras there are practical reasons for excluding the 1-dimensional case.
But I’m asking a more general theoretical question (insofar as there should be a general “theory” of simple objects in categories).
I went ahead and changed the definition, replacing zero object with terminal object. This allows the notion to capture more examples, for example simple rings. The fact that it doesn’t quite capture simple Lie algebras (because of a conventional fiat that simple Lie algebras are supposed to be nonabelian) seems more like a historical accident in naming conventions, than for any particularly great mathematical reason. Correct me if I’m wrong about that.
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